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1 Introduction1

1.1 Prologue:Algebra and Algorithms1

1.2 Motivations4

1.2.1 Constructive Algebra5

1.2.2 Algorithmic and Computational Algebra6

1.2.3 Symbolic Computation7

1.2.4 Applications9

1.3 Algorithmic Notations13

1.3.1 Data Structures13

1.3.2 Control Structures15

1.4 Epilogue18

Bibliographic Notes20

2 Algebraic Preliminaries23

2.1 Introduction to Rings and Ideals23

2.1.1 Rings and Ideals26

2.1.2 Homomorphism,Contraction and Extension31

2.1.3 Ideal Operations33

2.2 Polynomial Rings35

2.2.1 Dickson's Lemma36

2.2.2 Admissible Orderings on Power Products39

2.3 Gr？bner Bases44

2.3.1 Gr？bner Bases in K[x1,x2,...,xn]46

2.3.2 Hilbert's Basis Theorem47

2.3.3 Finite Gr？bner Bases49

2.4 Modules and Syzygies49

2.5 S-Polynomials55

Problems60

Solutions to Selected Problems63

Bibliographic Notes69

3 Computational Ideal Theory71

3.1 Introduction71

3.2 Strongly Computable Ring72

3.2.1 Example:Computable Field73

3.2.2 Example:Ring of Integers76

3.3 Head Reductions and Gr？bner Bases80

3.3.1 Algorithm to Compute Head Reduction83

3.3.2 Algorithm to Compute Gr？bner Bases84

3.4 Detachability Computation87

3.4.1 Expressing with the Gr？bner Basis88

3.4.2 Detachability92

3.5 Syzygy Computation93

3.5.1 Syzygy of a Gr？bner Basis:Special Case93

3.5.2 Syzygy of a Set:General Case98

3.6 Hilbert's Basis Theorem:Revisited102

3.7 Applications of Gr？bner Bases Algorithms103

3.7.1 Membership103

3.7.2 Congruence,Subideal and Ideal Equality103

3.7.3 Sum and Product104

3.7.4 Intersection105

3.7.5 Quotient106

Problems108

Solutions to Selected Problems118

Bibliographic Notes130

4 Solving Systems of Polynomial Equations133

4.1 Introduction133

4.2 Triangular Set134

4.3 Some Algebraic Geometry138

4.3.1 Dimension of an Ideal141

4.3.2 Solvability:Hilbert's Nullstellensatz142

4.3.3 Finite Solvability145

4.4 Finding the Zeros149

Problems152

Solutions to Selected Problems157

Bibliographic Notes165

5 Characteristic Sets167

5.1 Introduction167

5.2 Pseudodivision and Successive Pseudodivision168

5.3 Characteristic Sets171

5.4 Properties of Characteristic Sets176

5.5 Wu-Ritt Process178

5.6 Computation181

5.7 Geometric Theorem Proving186

Problems189

Solutions to Selected Problems192

Bibliographic Notes196

6 An Algebraic Interlude199

6.1 Introduction199

6.2 Unique Factorization Domain199

6.3 Principal Ideal Domain207

6.4 Euclidean Domain208

6.5 Gauss Lemma211

6.6 Strongly Computable Euclidean Domains212

Problems216

Solutions to Selected Problems220

Bibliographic Notes223

7 Resultants and Subresultants225

7.1 Introduction225

7.2 Resultants227

7.3 Homomorphisms and Resultants232

7.3.1 Evaluation Homomorphism234

7.4 Repeated Factors in Polynomials and Discriminants238

7.5 Determinant Polynomial241

7.5.1 Pseudodivision:Revisited244

7.5.2 Homomorphism and Pseudoremainder246

7.6 Polynomial Remainder Sequences247

7.7 Subresultants250

7.7.1 Subresultants and Common Divisors255

7.8 Homomorphisms and Subresultants262

7.9 Subresultant Chain265

7.10 Subresultant Chain Theorem274

7.10.1 Habicht's Theorem274

7.10.2 Evaluation Homomorphisms276

7.10.3 Subresultant Chain Theorem279

Problems283

Solutions to Selected Problems291

Bibliographic Notes296

8 Real Algebra297

8.1 Introduction297

8.2 Real Closed Fields298

8.3 Bounds on the Roots306

8.4 Sturm's Theorem309

8.5 Real Algebraic Numbers315

8.5.1 Real Algebraic Number Field316

8.5.2 Root Separation，Thom's Lemma and Representation319

8.6 Real Geometry333

8.6.1 Real Algebraic Sets337

8.6.2 Delineability339

8.6.3 Tarski-Seidenberg Theorem345

8.6.4 Representation and Decomposition of Semialgebraic Sets347

8.6.5 Cylindrical Algebraic Decomposition348

8.6.6 Tarski Geometry354

Problems361

Solutions to Selected Problems372

Bibliographic Notes381

Appendix A:Matrix Algebra385

A.1 Matrices385

A.2 Determinant386

A.3 Linear Equations388

Bibliography391

Index409